Question: Simplify; express your answer in exponential form. Assume $z\neq 0, y\neq 0$. $\dfrac{{(z^{4}y)^{-2}}}{{(z^{5}y^{3})^{3}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{4}y)^{-2} = (z^{4})^{-2}(y)^{-2}}$ On the left, we have ${z^{4}}$ to the exponent ${-2}$ . Now ${4 \times -2 = -8}$ , so ${(z^{4})^{-2} = z^{-8}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{4}y)^{-2}}}{{(z^{5}y^{3})^{3}}} = \dfrac{{z^{-8}y^{-2}}}{{z^{15}y^{9}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-8}y^{-2}}}{{z^{15}y^{9}}} = \dfrac{{z^{-8}}}{{z^{15}}} \cdot \dfrac{{y^{-2}}}{{y^{9}}} = z^{{-8} - {15}} \cdot y^{{-2} - {9}} = z^{-23}y^{-11}$